A Special Sequence - For my Friend Sharky Kesa!

A sequence of numbers \(x_{1},x_{2},\ldots ,x_{100}\) has the property that for every integer \(k\) between 1 and 100 inclusive the number \(x_{k}\) is \(k\) less than the sum of other 99 numbers. Given that \(x_{50}=\dfrac mn\), where \(m\) and \(n\) are relatively prime positive integer, find \(m+n\).